physical ageing of amorphous polymers. theoretical analysis and experiments on poly(methyl...

Makromol. Chem. 192, 2141 -2161 (1991) 2141

Physical ageing of amorphous polymers. Theoretical analysis and experiments on poly(methy1 methacrylate)

Joseph PPrez

Laboratoire GEMPPM, U.A. CNRS 341, INSA, 69621 Villeurbanne Cedex, France

Jean Yves Cavaille

CERMAV-CNRS, BP 53 X, 38041 Grenoble, France

Ricardo Diaz Calleja: JosP Luis Gomez Ribelles, Manuel Monledn Pradas, Amparo Ribes Greus

Laboratory of thermodynamics and physical chemistry, ETSII, Universidad Politecnica de Valencia, p. 0. box 22012, 46071 Valencia, Spain

(Date of receipt: June 13, 1990)a)

SUMMARY: Structural relaxation of glassy poly(methy1 methacrylate) (PMMA) was studied with several

techniques: differential scanning calorimetry (DSC), volumetry by thermomechanical analysis (TMA) and mechanical spectroscopy. A theory is proposed in order to interpret the data in a unique theoretical framework. This theory is developed on the basis of two main physical assumptions: i) the existence in amorphous polymers of quasipunctual defects whose concentration decreases during structural relaxation, and ii) the occurrence of hierarchical correlation effects conditioning molecular mobility. As a first approximation a self-consistent semiquantita- tive description of the results is obtained, i. e., the experimental results obtained for the different quantities on PMMA can be accounted for employing always the same set of model parameters. However, it seems that a significant difference exists between the relaxation times associated with the different experimental techniques.

Introduction

Structural recovery takes place in glass-forming materials during isothermal annealing at temperatures below the glass transition. Primary thermodynamical properties such as volume, v, or enthalpy, h, have been mainly considered for the study of this phenomenon. However, many other properties of amorphous materials can be affected by ageing. For example, dynamic mechanical and dielectrical properties are also influenced by the isothermal approach to equilibrium in such a way that the loss tangent diminishes, the storage modulus increases and the dielectric permitivity diminishes.

The purpose of this paper is to study the kinetics of the approach to equilibrium of the specific enthalpy and volume and their derivatives (the linear expansion coefficient and the specific heat) and of the dynamic mechanical properties of poly(methy1 meth-

a) Revised version of December 11, 1990.

C 1991, Huthig & Wepf Verlag, Basel CCC 0025-1 16X/91/$05.00

2142 J. Perez et al.

acrylate) (PMMA), and to apply a model based on physical considerations to reproduce the behavior observed experimentally in this polymer near the glass transition temperature Tg = 395 K. Finally, we analyze the differences observed between the kinetics associated to the recovery of each physical property.

Theory

Structural recovery in glass-forming systems has attracted considerable interest in the last years. After the pioneering works reviewed in ref.]), it was obvious that the thermodynamical properties h (enthalpy) and v (volume) were mainly to be considered during ageing and structural recovery of glasses. Furthermore, it was assumed that any non-equilibrium state of the glass could be characterized by the fictive temperature, T, . Thus, the time evolution of Tf following a temperature step AT from equilibrium (temperature To and time t = 0) is described by a relaxation function @ ( t ) as:

Experimentally, @ ( t ) appears to be (i) non-exponential and (ii) non-linear. The non-exponential aspect has been accounted for by using either a distribution of

relaxation times g(r), related to @ ( t ) through

with 00 I g(r) d7 = 1 0

or the so-called Williams-Watts (or Kohlrausch) function

An example of the former is the two-box distribution function proposed by Kovacs examples of the latter are the works of Narayanaswamy” and of Moynihan et al.

et in which the parameter r* is given by

Xh (1 - X ) h

RTf (4)

where A, 0 < X < 1 and h are constants, and R is the ideal gas constant.

of physical ageing of various glasses ‘ ~ ’ 3 ~ ) .

Adam-Gibbs theory*)

Both Eqs. (2) and (3) lead to multiparameter models and have been used in the study

Hodge’) has recently proposed the following expression for 5 resulting from the

Physical ageing of amorphous polymers . 2143

with

p is the free-energy barrier hindering cooperative rearrangement; S: is the configurational entropy of the smallest group able to rearrange; Acp is the configurational heat capacity; T, is the configurational ground-state temperature.

Thus, a link appears between the phenomenological parameters, X, h, p, r, and T, . In Eqs. (4) and ( 5 ) the non-linearity of the structural relaxation is introduced by means of the dependence of T on Tf. The parameter X in Eq. (4) is a numerical measure of the non-linearity ( X = 1 is expected for a linear relation). The main difference between Eqs. (4) and ( 5 ) is the predicted dependence of T on the temperature in equilibrium (when Tf = T): Eq. ( 5 ) predicts the well known Fulcher-Thmmann-Vogel (FTV) dependence instead of the Arrhenius equation predicted by model (4).

Several points can be noted in connection with these models: a) A choice of the appropriate set of parameters allows to reproduce reasonably well

the experimental curves measured after a fixed thermal history. Nevertheless, a single set of parameters is not able to reproduce the curves measured after different thermal histories 9, lo).

b) The FTV curve predicted by Eq. (5) for log[z(T)] in equilibrium can be approximated by a straight line in the range of temperatures between r, and

- 50 "C (the one used in our experiments) with an accuracy higher than that of the fit of the experimental results. The straight line predicted by Eq. (4) and the FTV curve predicted by Eq. ( 5 ) are very close to each other. In fact, they coincide within the accuracy of the fit. Of course, in the case of Eq. (4) the value of the preexponential factor A has no physical meaning. This equation should be considered as an approximation valid only in the range of temperatures of the ageing experiments but not at temperatures higher than q .

c) An inverse correlation is found between X and h in Eq. (4) and between D and T2 in Eq. (5 ) . The physical interpretation of such a correlation is obscure.

d) The choice of the distribution of relaxation times (or of the value of /3) is rather arbitrary among several possibilities "); the determination of this distribution is generally made as for a fitting parameter but its physical content is ignored.

As an alternative, we intend to develop a new approach in order to interpret on a physical basis the relaxation phenomena observed in glasses. Parts of this theory have been published recently I,), and only the main features will be recalled here.

a) As a first assumption, we propose to consider the occurrence in the subcooled liquid state of micro-density fluctuations which are frozen in at T < q; such fluctuations in density (i. e., in enthalpy and entropy) are called hereafter quasi- punctual defects.

By the usual arguments of statistical thermodynamics, the defect concentration is given by 13)


and

where A H F ( i ) is the enthalpy due to broken van der Waals bonds and excited intramolecular bonds, and AS,( i ) is the entropy increment related to the different possible states of intramolecular bonds; 1 < i < Nt , and Nt is the number of values of AH# and ASF(i ) proposed in order to describe the microstructure of the polymer. The defects presented here are merely sites with density fluctuations. Moreover, such fluctuations may be either positive or negative, similar to the p and n defects, respectively, previously introduced by Slorovitz et al. 14) for metallic glasses. But in all cases a defective site implies a positive fluctuation of enthalpy (and entropy). It is however worthwhile to notice that negative density fluctuations can be compared to free volume: nevertheless, the concept of defect might be more useful for at least two reasons:

(i) it contains both free volume and anti-free volume concepts, and (ii) it takes into account the interaction between structural units, which is strong in

condensed matter such as non-crystalline materials near Tg , via enthalpy fluctuations. Thus, at T > Tg there are density, enthalpy and entropy fluctuations, and repeating units are alternatively either closepacked, or they form with their neighbours disordered regions or defects. At T < T, these defects are frozen in, leading to local zones of high molecular mobility, which might be similar to the islands of mobility introduced by Johari ‘ 5 ) .

b) The diffusion process results from atomic (molecular) movements presenting strong correlation effects; furthermore, we assume that these correlation effects are hierarchical, implying a series distribution of characteristic times. This leads 1 6 ) to a new expression for the time of a molecular movement:

I / b

Tmol = (+) 7, is a characteristic time for the elementary molecular movement of the main chain

supposed to occur more rapidly in quasi-punctual defects. In the case of polymer materials, it can be considered that rl is connected with the degree of intermolecular rotational freedom of the smallest subchain atoms along the chain. Such a possibility is well known as a “crankshaft” process, and we suggest to identify rl with r B , the characteristic time corresponding to the fl relaxation. Hence, we shall use:

to = to . exp (+) (9)

Physical ageing of amorphous polymers . . . 2145

T~ and U, are, respectively, the limit time and the activation energy of the process. Furthermore, in Eq. (8) T,, < to < T~ is a parameter fixing the time scale of the whole correlated event, and 0 < b < 1 is a parameter characterizing the correlation between the different atomic movements: the stronger the interaction between structural units (i. e., the lower the temperature and, consequently, the lower the concentration of quasi- punctual defects), the lower is the value of b. We propose as a first approximation to relate the parameter b linearly to the total concentration of defects, C,

where the constant C, lies’,) around 3. c) The basic mechanism for non-elastic deformation of amorphous solids was

described earlier ‘ 3 3 16,17) in terms of nucleation of shear microdomains. The rate of such nucleation was previously calculated by several authors, but the results appeared to be unrealistic as the nucleation can occur only in those regions where resistance to shear is appreciably weaker than in the rest of the material. Such soft sites may be regarded as those defects evoked above. The thermomechanical activation of a defect (mean time T ~ ) may lead to the formation of a shear microdomain. When the stress is removed, the solid recovers its previous configuration, and this corresponds to the anelastic behaviour. In order to obtain plastic deformation, the growth of the shear microdomains must be necessarily invoked. Nevertheless, the line bordering the sheared area, which is defined as a dislocation loop in the mechanics of continuous media, is a non-glissile defect in amorphous solids (Somigilana type); hence, micro-domain growth is only possible through a diffusional mechanism: this growth covers a distance (mean time r2) at which the line bordering the sheared area loses its identity by combination with other similar lines formed from the neighbouring defects, and at which it becomes ineffective (viscoplastic behaviour).

Then, the number of activated defects n ( t ) (i. e., entering a shear microdomain) can be calculated as a function of time; it has been shown that 5 , and T, are both related 16, 1 7 ) to T,,, . Such a calculation leads to the global compliance; taking the Fourier transform and introducing the rubber elasticity, one obtains the complex modulus

with 5, = C, * T,,, the mechanical relaxation time for the main or a relaxation in amorphous polymers; H a numerical parameter close to 1; 0 < b’ < 1 a parameter indicating the sensibility of b on the deformation (b’ > 0,9 for polymers having long subchain segments free between entanglements; b’ < 0,5 for densely crosslinked polymers).

To sum up, in this theory volume and enthalpy recovery experiments as well as dynamic-modulus changes can be described as a consequence of the variation with temperature or time of the concentration of quasi-punctual defects: hence, C, appears to be one of the main ingredients of the theory.


Thanks to the points (a), (b) and (c), it is possible to describe the changes induced by structural relaxation in properties such as enthalpy, volume and dynamic modulus, as explained in what follows.

d) In the supercooled liquid, defects result from thermal fluctuations transforming normal sites (NS) into defects, and the following equilibrium equation could be used

(12)

with d + and d - referring to positive and negative density fluctuations, respectively. One has a set of Nt equations:

2

1 2NS d f + d - + AG

2NS d l + dT + AG; (13)

Actually, the situation is much more complicated as other equations such as

d t + d; dk+ + d; + AG(k10) (14)

must also be taken into account, with a relation between the integers i, j , k and I (these integers lie between 0 and Nt , the value 0 corresponding to a normal site).

To work with the whole set of equations (14) is an overwhelming task. We suggest as an approximation the use of Eqs. (1 3). Thus, direct and reverse structural relaxations correspond to directions and respectively, of the equilibrium described by those equations. In both cases, the kinetics of evolution is governed by the diffusion of defects, and the scale for diffusion length can be identified with the mean distance I, between defects i. Then, we have the equation

d C i C;(t) - C i ( 0 ~ ) (15)

This is a phenomenological equation, but it is possible to introduce parameters having a definite physical meaning. So, the structural relaxation time T~~ can be given by:

(1 6)

We now introduce D i , the coefficient of diffusion of defects i, which can be

approximated by -, where 1 is the mean length of elementary jumps of defects

(similar to the single structural unit, or monomer, dimension) and r;, the mean duration of those jumps, is identified with r,,, (see Eq. (8)).

- - - d t Tsr

T,, = 1; * D-'

1 2 r;

Assuming that

Eqs. (15), (16) and (17) lead to

Physical ageing of amorphous polymers . . .

with i / b ( l )

51 = (-) . Ci(t)-2’3

2147

(19)

and b depending on t in the same way as Cd(t) (Eq. (10)). Recalling that Cd(t) = C,( t ) , we have the global equation

i

It can be verified that the set of Eqs. (17)-(20) is equivalent to Eqs. (1)-(5). However, the analysis presented here gives a clear physical insight to both the non- exponential and non-linear character of structural relaxation, describing them in terms of defect concentration changes: the former results from the distributed properties of defects, implying distributed values in their mobility; the latter is due to the change in correlation effects when the defect concentration varies.

Eq. (1 8) can obviously be used for isothermal experiments; by approximating a linear increase of temperature as a series of isothermal steps, the temperature dependence of properties linked to the defect concentration can be obtained: for instance, the rate of heating (or cooling) heat capacity, cp , the expansion coefficient, a, , and the dynamic modulus.

Let us see how the variation of the different properties can be calculated. For enthalpy or, more precisely stated, for the excess configurational enthalpy AH,

the situation is clear. We have per mole of material:

To establish a relationship between Ci or c d and the specific volume is less obvious. When a defect d t is annihilated with a defect d; , a change of volume can be expected due to anharmonicity; such a feature is to be compared with the phonon thermal expansion coefficient. We consider this thermal expansion coefficient of the glass:

A V V

Eq. (22) means that the relative volume expansion - results from a change AT

of temperature inducing an increase kB . A T of the potential enthalpy of interaction between neighbouring units. When creation (or annihilation) of defects is concerned, the change of enthalpy is given by AHF(i). Thus, the increment of volume per mole due to defects in thermodynamic equilibrium can be derived from Eq. (22) as

Finally, if the relative change of excess volume, hereafter called d(t ) , is defined by


Eq. (23) leads to:

Having C,(t, T ) , i. e., b(t, T ) (Eqs. (20) and (lo)), the dynamical modulus as well as the enthalpy and the specific volume can be calculated as a function either of time or of temperature with Eqs. (1 l ) , (21) and (24).

A simulation program has been developed in order to calculate the above mentioned properties for the case of PMMA in several thermal histories.

Computer simulation - comparison with experiments

As explained above, it is possible to make a computer simulation of the behaviour of an amorphous polymer during ageing. A well known material has been chosen, PMMA, and three properties have been studied: specific volume, enthalpy and dynamical modulus. The challenge was not to obtain the best fit between the experimental data and the calculated curves for all properties (although this is possible in the case of each property), but rather to obtain a self-consistent simulation of the whole behaviour by using a unique set of parameters. A reasonable (or even good) agreement with experimental data is obtained.

All this can be used to deduce information about the molecular aspect of the structural relaxation. In a previous work Is), a similar attempt was made but, on one hand, the best way to determine the parameters was not established, and, on the other hand, a greater number of experimental data is available now.

Regarding the determination of the parameters, there are 3 groups of them: (i) those related to the thermodynamic state of the material, AHF and ASF (and

(ii) those corresponding to dynamical properties, t o , b (Eq. (S)), U, , r,, (Eq. (9)), C ,

(iii) those related to the modulus: C,, G, , G,, b' (Eq. (1 1)). In short, the parameters of the two latter groups are obtained from mechanical

spectrometry 1 6 , "1, and those of the former group can be deduced from calorimetric experiments, as will be shown in what follows.

consequently C, ( T J ) ,

(Eq. (lo)), and

Experimental part

Calorimetry

A sample of 6 mg of polymer was sealed in an aluminium pan and employed in all the DSC measurements. A Perkin-Elmer DSC4 differential scanning calorimeter with a data station model 3 600 was used. Two different types of experiments were performed: in the first one, the sample was annealed at 150 "C for 10 min to ensure that it was in equilibrium and so to erase the effects of previous thermal histories. The sample was then cooled down with a rate of 1 0 " C h i n to the temperature T, (aging temperature), kept at this temperature for a time t , and cooled down again


with 10"C/min until T, = 40"C, the temperature selected for the start of the measuring scan, which took place while heating with a rate of lO"C/min until 150°C. The measuring scan was then repeated with an empty pan in place of the sample pan. From these two measurements, the specific heat of the polymer, c,(T) was calculated between 60 and 150 "C. The calorimeter was frequently calibrated with a sapphire standard.

The derivative of the fictive temperature with respect to the temperature, dTf/dT, was determined using Eq. (4)

cpl being the specific heat corresponding to equilibrium (liquid state), cPg the one corresponding to the glassy state and cP,exp the specific heat measured after a given thermal history. T* is a temperature high enough above T g . Thus, the derivative dTf/dT is equivalent to a normalized specific heat, since in equilibrium Tf/dT = 1, and in the glassy state dTf/dT = 0.

A second type of experiments was performed with the aim of measuring the increment of specific enthalpy of the sample during the isothermal step at T,, called the enthalpy lost due to ageing, ha . The method of Lagasse 19) was used: the sample was annealed at 150 "C for 10 min, cooled down with 10 "C/min to T, , kept at this temperature during t,, and then the measuring scan was carried out between T, and 150°C with 10"C/min. Another measuring scan followed with t , = 0. The difference in heat flux between both scans allows the calculation of ha.

Poly(methy1 methacrylate) (PMMA) samples for all kinds of tests (calorimetric, dilatometric and dynamic mechanical) were prepared by radical bulk polymerization of methyl methacrylate (from Merck) at 60°C for 14 h, with 0,06 wt.-% of 2,2'-azoisobutyronitrile as initiator. The samples were then annealed at 90 "C for 4 h to complete the polymerization and dried at 70 "C i. vac. until their weight remained constant. Typical weight-average molecular weights obtained for these kinds of samples lie between 1 , O . lo6 and 1,2. lo6 in all cases.

Dilatometry

Two types of experiment were conducted in a TMA 943 Dupont apparatus. A first group of tests consists of the measurement of the change in length during annealing at

a constant temperature after quenching with - 10 "C/min from above Tg to the annealing temperature.

In the second type of tests, the sample was quenched and annealed in the same conditions as above; then it was again quenched to room temperature, from which temperature it was heated to temperatures above Tg (160- 170°C) whilst measuring its change of length. The ageing times were 10, 30, lo2, 3 . lo2, lo3 and 3 . lo3 min, and the heating rate was 5 "C/min.

The first type of tests determines the isothermal approach of the length to its equilibrium value corresponding to the fixed temperature; the second type of tests delivers the curve of the change in length as a function of temperature during the heating scan, as affected by the ageing process.

For this type of test the sample had the shape of a prismatic bar of dimensions 1,93 x 6,30 x 6,45 mm. The values of the linear expansion coefficient determined experimentally were converted to the volume expansion coefficient by multiplication by a factor 3.

Dynamic mechanical spectrometry

A forced torsion pendulum, Mechanalyser from Metravib Instruments, was used to measure the storage modulus and the loss tangent in a range from to 1 Hz, from 150 to 400 K. The rates of cooling and heating were, respectively, - 12 "C/rnin and 0,6 "C/min. For these tests the samples were prismatic bars of dimensions 1 x 5 x 50 mm.


Results and discussion

CaIorimetry

The computer simulation of the c,(T) curves was made by using

AHF(i) = AHF(0) + d H . i

with 1 Q i < 20, and

where

So only dHhas to be chosen to determine the distribution width in the characteristics of defects. For each value of dH, AH, (0) and ASF (0) are obtained from the following equations

with 8, To = 0,8 Tg ,

The value of Acp was determined from unaged scans. A mean value of 0,268 J/(g * K) was used in the computer simulation.

The solution of Eqs. (29) and (30) can be obtained with a computer calculation based on Newton's method. For instance, with dH = 0 (no distribution) we obtain AHF(0) = 18,6 kJ/mol and ALS,(O) = 5,2 J/(mol * K), implying C,(T,) = C, (395 K) = 0,12.

Parameters connected to the dynamic properties are determined, as will be explained later, from mechanical spectrometry. The following values have been used: T~ = 5 s, U p = 69 kJ/mol (p relaxation); b = 0,28 near Tg , leading to C, = 2,5 according to the relation between C, and b.

Calorimetric experimental results obtained after several thermal histories are represented in curves of dT,/dT in Figs. 1 a and 2a. The curve corresponding to the unaged sample is represented in both figures for the sake of comparison. The peak observed in the case of the sample aged at 103 "C overlaps with the one appearing in the unaged sample. This behaviour is characteristic of an ageing at a temperature close to Tg . After ageing at 80 "C, the peak resulting from the ageing treatment appears in an intermediate position, lower than the glass transition range; the high-temperature part of the curve is, then, practically the same as in the unaged sample. In Figs. 1 b and 2 b the results predicted by the model with to = 1,5 * s are represented. The agreement with the experimental results is not only qualitative, but also quite apparent in the position and the height of the peak in all the thermal histories tested.

Physical ageing of amorphous polymers . . 2151

h U \ L'

u 2

1

0

L -0 \ k - 2

1

0

100 150 Temp. in O C


Fig. 1 . (a) Experimental variation of dT,/dTwith temperature, and (b) calculated curves, after different ageing times at 103 "C: t , = 0 (-), t , = lo2 min (. . . . . .), t , = lo3 min (- - -)

L U \ L- -0

1

h -0 \ L-

(a)

-0

(bl



Fig. 2. (a) Experimental variation of dTf/dT with temperature after different ageing times at 80 "C. (b) Calculated curves. t , = 0 (-), t , = 3 . 10' min (. . . . . -), t , = 1273 min (- - -)

2152 J. PCrez et al.

The measurement of the increment of enthalpy in the isothermal step of the thermal history, h a , was performed following the method proposed by Lagasse, as described in the Exptl. part. The curves of ha as a function of the ageing time at different ageing temperatures are shown in Fig. 3. Due to the difficulty of defining precisely the experimental beginning of the ageing ( t , = 0), preventing the simulation of the whole experiment (cooling, isothermal ageing and heating) we propose to compare the experimental data and the theory with the aid of the relaxation function @ ( t ) (Eqs. (1)-(3) and (20)).

0.5

0

Fig. 3. Released enthalpy as a function of time for different ageing temperatures: (0) 115 OC, (0) 112,5'C, ( 0 ) llO"C, (A) 105"C, (A) 103"C, (V) 93"C, and straight line of slope 2AcP (see Eq. (42))

Let us consider the isothermal recovery process which, at temperature T,, takes the material from a state out of equilibrium in which the specific enthalpy is h (T,, 0) to the equilibrium state in which the enthalpy is heq(Ta).

This process can be described by means of a decay function

where

h(T,,t,) - heq(Ta) = h(T,,O) - heq(T,) - ha (32)


It is necessary to evaluate the term h (T , , 0) - heq (T,) in order to transform our experimental data of h,(T, , I,) into @(Ta , t,) curves. This is quite easy when T, is higher than 112,5 "C, because a plateau zone in the curve of ha is attained, so:

lim ha = h(T,,O) - heq(T,) fa-"

(34)

At temperatures below 112,5 "C the accuracy of the value of h(T,,O) - heq(Ta) determined by extrapolation is very poor and a different procedure has to be used. At low temperatures, in the glassy state, the states reached by cooling from the liquid state can be characterized by means of a value of Tf independent of temperature but dependent on the cooling rate. Thus, at low temperatures

Tf(T,,O) = Ti (35)

The value of Ti calculated for a cooling rate of 10"C/min is 114,7 f 0,2 "C. If, as an approximation, an average value of Ac, = 0,268 J/g is assumed in the range of temperatures of the glass transition,

(36) h

- q(Ta,ta) = Tf(Ta,O) - T,(Ta,ta) = a ACP

At ageing temperatures below 112,5 "C the relaxation function was calculated as

Fig. 4 shows the values of cp (Ta , t,) for the different ageing temperatures. Let US

recall the well known Davies and Jones equation"), which corresponds to Eq. ( I 5):

Fig. 4. Relaxation function corresponding to the data of Fig. 3: Experi- mental data and theoretical curves (full lines). Temperatures: (0) 115"C, (M) 112,5"C, (V) l l O T , ( 0 ) 105% (n) I O ~ T , (A) 9 3 ~

1 2 3

2154

T f - Ta - -~ d =f

dta TSI

_ _ -

After integration we have

with, similarly to Eq. (19),

and

J. PCrez et al.

(38)

Eqs. (39)-(41) have been solved numerically. The value of tdadjusted to fit the data was 6,5. s and a Gaussian distribution (coefficient equal to 5 ) for T,, was assumed. Theoretical curves are drawn in Fig. 4. A good fit is observed and the tendency of the data to merge at short t, is taken into account. Furthermore, such a description takes into account the important experimental fact in describing annealing effects, i. e., the correspondence of a change of 2 to - 3 K in the value of Tf for one decade in time near T,(O). To show this, as does Bauwens2'), a straight line is drawn in Fig. 3, the slope of which is equal to

1 dAH

Acp dlogt, -.-- - 2

To sum up the information given in Figs. 3 and 4, it is worthwhile to notice that Eqs. (38) to (41) (especially Eq. (40)) can be used successfully to describe the effect of temperature on the relaxation kinetics: not only the form of the curves is correctly obtained but also the effect of temperature of annealing is accurately reproduced.

Dilatometry

The experimental curves after subtraction of ag are given in Figs. 5a and 6a, respectively, for two ageing temperatures (366 K and 376 K). Computer calculations of volume changes are shown by comparison between Eqs. (21) and (22), (23) and (24). The only new parameter to be introduced in the calculations is the thermal expansion coefficient of glassy PMMA, u g , the value of which is (2,8 f 0,l). K - ' , so with the help of the Eqs. (20), (23) and (24) theoretical curves of A a ( T ) (= a ( T ) - a g ) have been drawn. Such curves correspond to non-annealed and annealed states at Ta =

366 K and 376 K (Figs. 5 b and 6 b, respectively). Calculations were made with the same ageing times as in the experiments.

A reasonable agreement is observed between the theoretical value of Aa = a, - ag = 3,3 * K - I . This can be viewed as an argument in favour of the validity of Eq. (22). Moreover, the height of the peaks

K - ' and the experimental one (2,8 f 0,l) *

Physical ageing of amorphous polymers 2155

Y

(a) Fig. 5 . (a) Experimental f 6 1 variation of the thermal Frn 5 volume expansion coefficient ha with temperature after different ageing times. (b) Calculated 3

- i: I , . . p..)

curve times: at t , 103 = 0 'C. (-), Ageing -i" 1 ,s'/ t , = 100min( . . . . . . ) , 0 t , = to3 min (- - -)

100 120 110 Temp. in O C

Y

? 6 - > 7 -

- 7 5 - - 1 -

3 - 2 -

1 -

0

100 120 110 Temp. in OC

variation of the thermal - Fig. 6 . (a) Experimental

volume expansion coeffi- tJ" cient Aa with tempera- U

ture after different ageing times. (b) Calculated curve at 93 "C. Ageing times: t , = 0 (- .- .-), f a = 100 min (-), ta = 300min( . . . . . . ) , t , = 3.103min(- - - )

- 1

100 120 110 100 120 110 Temp. in O C Temp. in O C

in Aa ( T ) varies with thermal history in a similar way in both types of curve and a good agreement is observed between the maximum height 6 - (experimental) and 7 . (calculated). In a general manner, the height and temperature of this peak increases with ageing times, and for the 366 K curve some evidence of a prepeak appears at shorter times, as reported by other authors.

Dynamic mechanical spectrometry

Shear storage modulus and loss tangent are shown in Fig. 7 a and 7 b, respectively, for a quenched sample and for a sample aged at 376 K for 7,2. lo3 min, each at frequencies of 0,l and 1 Hz.

2156

0 10 a > c3 - m 0 -

9

8

7

6

5

J. Perez et at.

11

I0

9

a

Fig. 7. quenching, measured at O, i Hz (. . . . . .) andat 1 Hz( - . - . ) , or ageing 7,2. lo3 min at 103°C. measured at 0,l Hz (-) and at 1 Hz (- - -); (b) idem for tan6; (c) corres-

(a) log G vs. T after

6 . ^ ponding, calculated tan 6 -98 2 102

Temp. in O L curves

0

-1

-2

2 102 Temp. in O C

0

-1

-2 I

Temp. in O C

3 2 102

Physical ageing of amorphous polymers . . . 21 57

To make an estimate of the relaxed and unrelaxed modulus, GR and G, , a G ' vs. G' Cole-Cole diagram is prepared after a superposition of the experimental curves at several frequencies and in order to plot master curves for both the storage and loss modulus.

The following values are obtained: G, = lo6 Pa, GI = 1,2 . lo9 Pa and H = 1, b' = 0,73, b = 0,28 (see Eq. (1 1)). The last parameter can be used to estimate, through Eq. (lo), the value of C, = 2,5. In order to complete the fit in the p relaxation region the equation

Gi - GR (43)

i w 'p (11 G * = c Sj + GR + , 1 + i w q ( j ) I + h(iw.r,)-b + (iw.r,)-'

is useful (see ref. 16) for details). In this equation values of T~ = 5 * IO-l3s, U , = 69 kJ/mol for the activation energy of the fi process are used. All these data are necessary in the calculation of tans from Eq. (43) and are valid for cooling or heating experiments thanks to Eq. (18). Calculations are carried out

i) by adjusting C, = T ~ / T , , , ~ , to 8 . in order to obtain good reproducibility in the position of the a peak and

ii) by using a Gaussian distribution of sP in Eq. (9) with a coefficient equal to 6 and cut-off for times higher than the most probable value of sP .

The results are shown in Fig. 7c. Once more the main experimental effects are reproduced, that is, (i) the loss tangent diminishes between the p and a relaxation, and (ii) this effect is more striking at lower frequencies.

Isothermal measurements were carried out on conditions which are comparable to those of the calorimetric and dilatometric experiments, T, = 376 K. Fig. 8a shows typical experimental results, from which it can be seen that the kinetics of change of tans and G' are the same. Thus, a unique normalized curve has been plotted in Fig. 8 b. which delivers both

tan 6 (fa) - tan 6 (OD)

tan6(0) - tand(w) 0tan*(t,) 5%

and G ( m ) - G(t,)

G ( m ) - G(0) @G'(ta) =

(44)

(45)

In Fig. 8b the calculated isothermal curve was obtained making use of Eqs. (1 7)-(20), as in the case of the enthalpy. The values of the parameters are the same as those for the calculated curves of Fig. 4 (to = 1,7 * 10-6s and C, = 2 . and Eq. (43) is used to obtain t ans during isothermal ageing.

Although the Eqs. (38) to (41) have been used both for enthalpy and for tan6 with the same values of the parameters, it is clear that the kinetics of tans is more rapid than the one for enthalpy and volume. A comparison between the kinetics of @(t , ) observed for enthalpy, volume and dynamic mechanical moduli in the same experimental conditions is given in Fig. 9. There, the sequence

7mechanical a relaxation < Tvolume < ?enthalpy

2158

K2 0.20 C 0 c

0.15

J. Perez et al.

3.5 - a" > c3 - m 0

9.0 -

8 5

8.0

7.5 1 2 3 4 5 6 0.10 '

log ( t , /s)

Fig. 8. (a) Isothermal change in G' and tans at 103 "C for 1 Hz as a function of ageing time. (b) Normalized curve of G' and t ans vs. ageing time at 103 "C for 1 Hz, measured (full line) and calculated (dotted line)

appears, in agreement with the conclusions of other author^^^^^^). This could be explained by assuming that only quasi-punctual defects having the largest enthalpy of formation are shear-source defects 24), these defects are the most mobile, resulting in shorter times to reach equilibrium. Or, in more classical terms, the fraction of the free volume existing in the glass that intervenes in the structural recovery for tan6 is less than the one for the enthalpy.

Physical ageing of amorphous polymers . . .

- -c I

0

1.0

0.5

21 59

-

-

1oL 01 10' 102 lo3

t/min

Fig. 9. volume and ( A ) enthalpy

Relaxation functions @ ( t ) versus ageing times at 103 "C for (0) dynamic modulus, (0)

This discrepancy in the time scale notwithstanding, the model accounts for features of the experimental behaviour such as the frequency dependence of the ageing effect. Fig. 10a shows tans data, rather than G data, as the former are (i) more sensitive to frequency changes than the latter and (ii) independent of the size of the sample. Calculated curves with the same values of the parameters as above are exhibited in Fig. lob.

Lo C 0 - 1.0

0.5

0

Lo lo) C

2 1.0

I 0 0 2 1

log (t,/rnin)

Fig. 10. 0,l Hz (- - -) and 1 Hz (- . - .). (b) Corresponding, calculated curves

(a) Variation of t ans with ageing time at 103 "C for the frequencies 0,01 Hz (-I,


The interpretation proposed here implies a variation of the correlation effect during ageing. Indeed, Eq. (10) indicates that b varies as C, during structural relaxation.

A confirmation of such a variation seems useful. So, G’ as a function of frequency has been measured in the a relaxation zone (Fig. 11). The maximum corresponds to wr = 1, so we have directly a value of r, in both ageing conditions at 103 “C, t , = 83 min and t , = 8,3 . lo3 min. Eqs. (8) and (1 1) show that the only function that can vary during isothermal measurements is b. A rough estimate of the variation of b after such an ageing gives a decrease of 0,02.

I

-3 -2 -1 log ( f /Hz)

Fig. 11. Loss modulus G versus frequency after two different ageing times, 83 min (0) and 8,3 . lo3 min (0)

The Spanish side acknowledges partial funding through the Spanish Government’s CZCYT under project MAT 88/0555.

’) A. J. Kovacs, J. M. Hutchinson, J. J. Aklonis, in “The structure of non crystalline materials’:

’) A. J. Kovacs, J. J. Aklonis, J. M. Hutchinson, A. R. Ramos, J. Polym. Sci., Polym. Phys. Ed.

3, 0. S. Narayanaswamy, J. Am. Ceram. Soc. 54, 491 (1971) 4, C. T. Moynihan, A. J. Esteal, M. A. Debolt, J. J. Tucker, J. Am. Ceram. Soc. 59, 12 (1976) ’) C. T. Moynihan, P. B. Macedo, C. J. Montrose, P. K. Gupta, M. A. Debolt, J. F. Dill, B. E.

Dom, P. W. Drake, A. J. Esteal, P. B. Elterman, R. Moeller, H. Sasabe, J. A. Wilder, in “The glass transition and the nature of the glassy state’: ed. by M. Goldstein, R. Simha, New York Acad. of Sci. 1976; p. 15 1. M. Hodge, in “Relaxations in complex systems’: ed. by K. L. Ngai, G. B. Wright, Office of Naval research, Arlington 1984, p. 65

ed. by P. H. Gaskell, Taylor & Francis, London 1977, p. 153

17, 1097 (1979)

’) I. M. Hodge, Macromolecules 20, 2897 (1981) *) G. Adams, J. M. Gibbs, J Chem. Phys. 43, 139 (1965)


‘) J. J. Tribone, J. M. O’Reilly, J. Greener, Macromolecules 19, 1732 (1986) lo) J. L. Gomez Ribelles, A. Ribes Greus, R. Diaz Calleja, Polymer 31, 223 (1990) ’ ’ ) D. Ng, J. J. Aklonis, in “Relaxations in complex systems’: ed. by K. L. Ngai, G. B. Wright,

1 2 ) J. Perez, Polymer 29, 483 (1988) 1 3 ) J. Perez, Rev. Phys. Appl. 21, 93 (1986) ‘ I ) D. Slorovitz, R. Maeda, V. Vitek, T. Ecami, Philos. Mag. A: 44, 847 (1981) I s ) G. P. Johari, in “Plastic deformation of amorphous and semicrystalline materials’: ed. by

1 6 ) J. Perez, J. Y. Cavaille, S. Etienne, C. Jourdan, Rev. Phys. Appl. 23, 125 (1988) 1 7 ) J. Y. Cavaille, J. Perez, B. P. Johari, Phys. Rev. B: 39, 241 1 (1989) Is) R. Diaz Calleja, J. Perez, J. L. Gomez Ribelles, A. Ribes Greus, Makromol. Chem., Mucromol.

1 9 ) R. R. Lagasse, J. Polym. Sci., Polym. Phys. Ed. 20, 279 (1982) ’O) R. 0. Davis, G. 0. Jones, Proc. R. Soc. London A: 217, 26 (1953) ”) C. Bauwens-Crowet, J X . Bauwens, Polymer 27, 709 (1986) ’*) S. E. B. Petrie, J. Polym. Sci., Polym. Phys. Ed. 10, 1255 (1972) 23) H. Sasabe, C. T. Moynihan, J. Polym. Sci., Polym. Phys. Ed. 16, 1447 (1978) 24) J. Perez, Acta Metall. 32, 21 33 (1984)

Office of Naval Research, Arlington 1984, p. 53

B. Escaig, C. G‘sell, Ed. de Phys., Les Houches 1982, p. 109

Symp. 27, 289 (1989)

physical ageing of amorphous polymers. theoretical analysis and experiments on poly(methyl...

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